Let’s solve each problem one by one, step by step.
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GROWTH FUNCTIONS
Formula:
> y = a(1 + r)^t
Where:
- a = initial amount
- r = growth rate (as a decimal)
- t = time in years
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Example 1: Population Growth
Given:
- Initial population (a) = 8,250
- Growth rate (r) = 3% per year → 0.03
- Time (t) = 10 years
Plug into formula:
y = 8250 × (1 + 0.03)^10
y = 8250 × (1.03)^10
Now calculate (1.03)^10:
Using calculator or known value:
(1.03)^10 ≈ 1.343916
Then:
y ≈ 8250 × 1.343916 ≈ ?
Let’s compute that:
8250 × 1.343916
First, 8000 × 1.343916 = 10,751.328
Then, 250 × 1.343916 = 335.979
Add them: 10,751.328 + 335.979 =
11,087.307
Round to nearest whole number (since population is people):
→
11,087
✔ Final Answer for Example 1:
11,087
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Example 2: Salary Increase
Given:
- Starting salary (a) = $24,000
- Increase rate (r) = 1.5% → 0.015
- Time (t) = 6 years
Formula:
y = 24000 × (1 + 0.015)^6
y = 24000 × (1.015)^6
Calculate (1.015)^6:
≈ 1.093443 (using calculator)
Then:
y ≈ 24000 × 1.093443 = ?
Compute:
24000 × 1.093443
= 24000 × 1 + 24000 × 0.093443
= 24000 + 2242.632
=
26,242.632
Round to nearest cent (money):
→
$26,242.63
✔ Final Answer for Example 2:
$26,242.63
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DECAY FUNCTIONS
Formula:
> y = a(1 - r)^t
Where:
- a = initial amount
- r = decay rate (as a decimal)
- t = time in years
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Example 1: Car Depreciation
Given:
- Initial value (a) = $16,000
- Decay rate (r) = 7% → 0.07
- Bought in 2012, want value in 2018 → t = 2018 - 2012 = 6 years
Formula:
y = 16000 × (1 - 0.07)^6
y = 16000 × (0.93)^6
Calculate (0.93)^6:
≈ 0.646990 (using calculator)
Then:
y ≈ 16000 × 0.646990 = ?
Compute:
16000 × 0.646990
= 16000 × 0.6 = 9600
16000 × 0.046990 = 751.84
Total = 9600 + 751.84 =
10,351.84
Wait — let me recalculate more accurately:
Actually:
16000 × 0.646990 =
Let’s do 16000 × 0.64699 =
Break it down:
10000 × 0.64699 = 6469.9
6000 × 0.64699 = 3881.94
Total = 6469.9 + 3881.94 =
10,351.84
✔ Final Answer for Example 1:
$10,351.84
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Example 2: Stock Investment Depreciation
Given:
- Initial investment (a) = $12,000
- Depreciation rate (r) = 3% → 0.03
- Time (t) = 8 years
Formula:
y = 12000 × (1 - 0.03)^8
y = 12000 × (0.97)^8
Calculate (0.97)^8:
≈ 0.783743 (using calculator)
Then:
y ≈ 12000 × 0.783743 = ?
Compute:
12000 × 0.783743
= 12000 × 0.7 = 8400
12000 × 0.083743 = 1004.916
Total = 8400 + 1004.916 =
9,404.916
More accurately:
12000 × 0.783743 =
Let’s multiply directly:
12000 × 0.783743 = 9404.916
Round to nearest cent:
→
$9,404.92
✔ Final Answer for Example 2:
$9,404.92
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Final Answer:
Example 1 (Population): 11,087
Example 2 (Salary): $26,242.63
Example 1 (Car): $10,351.84
Example 2 (Stock): $9,404.92
Parent Tip: Review the logic above to help your child master the concept of worksheet exponential growth and decay.
